LMFDB - Embedded newform 392.4.i.o.177.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,4,Mod(177,392)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("392.177");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	392.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.1287487223$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.54095201243136.19
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 102x^{6} + 320x^{5} + 4283x^{4} - 9104x^{3} - 85298x^{2} + 89904x + 714364 $ x^8 - 4x^7 - 102x^6 + 320x^5 + 4283x^4 - 9104x^3 - 85298x^2 + 89904*x + 714364
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			177.1
Root			$5.10797 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	392.177
Dual form			392.4.i.o.361.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.81898 - 8.34673i) q^{3} +(2.95919 - 5.12547i) q^{5} +(-32.9452 + 57.0628i) q^{9} +O(q^{10})$	\(q+(-4.81898 - 8.34673i) q^{3} +(2.95919 - 5.12547i) q^{5} +(-32.9452 + 57.0628i) q^{9} +(9.18493 + 15.9088i) q^{11} +90.9844 q^{13} -57.0412 q^{15} +(45.9619 + 79.6084i) q^{17} +(-62.1237 + 107.601i) q^{19} +(-32.5206 + 56.3273i) q^{23} +(44.9864 + 77.9187i) q^{25} +374.825 q^{27} +83.4110 q^{29} +(-82.6540 - 143.161i) q^{31} +(88.5240 - 153.328i) q^{33} +(-155.486 + 269.310i) q^{37} +(-438.452 - 759.421i) q^{39} -181.504 q^{41} +82.3699 q^{43} +(194.982 + 337.719i) q^{45} +(183.954 - 318.618i) q^{47} +(442.980 - 767.263i) q^{51} +(-26.3287 - 45.6026i) q^{53} +108.720 q^{55} +1197.49 q^{57} +(280.639 + 486.081i) q^{59} +(138.152 - 239.287i) q^{61} +(269.240 - 466.337i) q^{65} +(30.0684 + 52.0800i) q^{67} +626.865 q^{69} -155.534 q^{71} +(365.739 + 633.479i) q^{73} +(433.577 - 750.978i) q^{75} +(-145.192 + 251.480i) q^{79} +(-916.754 - 1587.86i) q^{81} +43.5791 q^{83} +544.041 q^{85} +(-401.956 - 696.209i) q^{87} +(-196.547 + 340.429i) q^{89} +(-796.617 + 1379.78i) q^{93} +(367.672 + 636.827i) q^{95} -521.748 q^{97} -1210.40 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 136 q^{9}+O(q^{10}) $ 8 * q - 136 * q^9	$ 8 q - 136 q^{9} + 116 q^{11} + 224 q^{15} + 80 q^{23} - 448 q^{25} + 72 q^{29} - 436 q^{37} - 2232 q^{39} + 744 q^{43} + 780 q^{51} - 976 q^{53} + 7624 q^{57} - 780 q^{65} + 1176 q^{67} - 4816 q^{71} - 56 q^{79} - 2104 q^{81} + 9880 q^{85} - 2376 q^{93} - 4032 q^{95} - 5176 q^{99}+O(q^{100}) $ 8 * q - 136 * q^9 + 116 * q^11 + 224 * q^15 + 80 * q^23 - 448 * q^25 + 72 * q^29 - 436 * q^37 - 2232 * q^39 + 744 * q^43 + 780 * q^51 - 976 * q^53 + 7624 * q^57 - 780 * q^65 + 1176 * q^67 - 4816 * q^71 - 56 * q^79 - 2104 * q^81 + 9880 * q^85 - 2376 * q^93 - 4032 * q^95 - 5176 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−4.81898	−	8.34673i	−0.927414	−	1.60633i	−0.787632	−	0.616146i	$-0.788693\pi$
$3$	−4.81898	−	8.34673i	−0.927414	−	1.60633i	−0.139782	−	0.990182i	$-0.544640\pi$
$4$		0			0
$5$	2.95919	−	5.12547i	0.264678	−	0.458436i	−0.702801	−	0.711386i	$-0.748068\pi$
$5$	2.95919	−	5.12547i	0.264678	−	0.458436i	0.967479	+	0.252951i	$0.0814010\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−32.9452	+	57.0628i	−1.22019	+	2.11344i
$10$		0			0
$11$	9.18493	+	15.9088i	0.251760	+	0.436061i	0.964010	−	0.265864i	$-0.0856573\pi$
$11$	9.18493	+	15.9088i	0.251760	+	0.436061i	−0.712250	+	0.701925i	$0.752324\pi$
$12$		0			0
$13$	90.9844			1.94112			0.970558	−	0.240866i	$-0.0774313\pi$
$13$	90.9844			1.94112			0.970558	+	0.240866i	$0.0774313\pi$
$14$		0			0
$15$	−57.0412			−0.981864
$16$		0			0
$17$	45.9619	+	79.6084i	0.655730	+	1.13576i	0.981710	+	0.190381i	$0.0609723\pi$
$17$	45.9619	+	79.6084i	0.655730	+	1.13576i	−0.325980	+	0.945377i	$0.605694\pi$
$18$		0			0
$19$	−62.1237	+	107.601i	−0.750114	+	1.29924i	0.197653	+	0.980272i	$0.436668\pi$
$19$	−62.1237	+	107.601i	−0.750114	+	1.29924i	−0.947767	+	0.318963i	$0.896665\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−32.5206	+	56.3273i	−0.294827	+	0.510655i	−0.974945	−	0.222448i	$-0.928595\pi$
$23$	−32.5206	+	56.3273i	−0.294827	+	0.510655i	0.680118	+	0.733103i	$0.261929\pi$
$24$		0			0
$25$	44.9864	+	77.9187i	0.359891	+	0.623350i
$26$		0			0
$27$	374.825			2.67167
$28$		0			0
$29$	83.4110			0.534105			0.267052	−	0.963682i	$-0.413950\pi$
$29$	83.4110			0.534105			0.267052	+	0.963682i	$0.413950\pi$
$30$		0			0
$31$	−82.6540	−	143.161i	−0.478874	−	0.829434i	0.520832	−	0.853659i	$-0.325622\pi$
$31$	−82.6540	−	143.161i	−0.478874	−	0.829434i	−0.999707	+	0.0242245i	$0.992288\pi$
$32$		0			0
$33$	88.5240	−	153.328i	0.466971	−	0.808818i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−155.486	+	269.310i	−0.690860	+	1.19660i	0.280697	+	0.959796i	$0.409434\pi$
$37$	−155.486	+	269.310i	−0.690860	+	1.19660i	−0.971557	+	0.236807i	$0.923899\pi$
$38$		0			0
$39$	−438.452	−	759.421i	−1.80022	−	3.11807i
$40$		0			0
$41$	−181.504			−0.691369			−0.345685	−	0.938351i	$-0.612353\pi$
$41$	−181.504			−0.691369			−0.345685	+	0.938351i	$0.612353\pi$
$42$		0			0
$43$	82.3699			0.292123			0.146061	−	0.989276i	$-0.453340\pi$
$43$	82.3699			0.292123			0.146061	+	0.989276i	$0.453340\pi$
$44$		0			0
$45$	194.982	+	337.719i	0.645917	+	1.11876i
$46$		0			0
$47$	183.954	−	318.618i	0.570904	−	0.988835i	−0.425569	−	0.904926i	$-0.639926\pi$
$47$	183.954	−	318.618i	0.570904	−	0.988835i	0.996473	−	0.0839092i	$-0.0267406\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	442.980	−	767.263i	1.21627	−	2.10663i
$52$		0			0
$53$	−26.3287	−	45.6026i	−0.0682363	−	0.118189i	0.829889	−	0.557929i	$-0.188404\pi$
$53$	−26.3287	−	45.6026i	−0.0682363	−	0.118189i	−0.898125	+	0.439740i	$0.855071\pi$
$54$		0			0
$55$	108.720			0.266541
$56$		0			0
$57$	1197.49			2.78266
$58$		0			0
$59$	280.639	+	486.081i	0.619256	+	1.07258i	0.989622	+	0.143696i	$0.0458989\pi$
$59$	280.639	+	486.081i	0.619256	+	1.07258i	−0.370366	+	0.928886i	$0.620768\pi$
$60$		0			0
$61$	138.152	−	239.287i	0.289977	−	0.502255i	−0.683827	−	0.729644i	$-0.739686\pi$
$61$	138.152	−	239.287i	0.289977	−	0.502255i	0.973804	+	0.227390i	$0.0730191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	269.240	−	466.337i	0.513771	−	0.889877i
$66$		0			0
$67$	30.0684	+	52.0800i	0.0548275	+	0.0949639i	0.892136	−	0.451766i	$-0.149206\pi$
$67$	30.0684	+	52.0800i	0.0548275	+	0.0949639i	−0.837309	+	0.546730i	$0.815872\pi$
$68$		0			0
$69$	626.865			1.09370
$70$		0			0
$71$	−155.534			−0.259979			−0.129989	−	0.991515i	$-0.541494\pi$
$71$	−155.534			−0.259979			−0.129989	+	0.991515i	$0.541494\pi$
$72$		0			0
$73$	365.739	+	633.479i	0.586391	+	1.01566i	0.994701	+	0.102815i	$0.0327849\pi$
$73$	365.739	+	633.479i	0.586391	+	1.01566i	−0.408310	+	0.912843i	$0.633882\pi$
$74$		0			0
$75$	433.577	−	750.978i	0.667536	−	1.15621i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−145.192	+	251.480i	−0.206777	+	0.358148i	−0.950697	−	0.310120i	$-0.899631\pi$
$79$	−145.192	+	251.480i	−0.206777	+	0.358148i	0.743921	+	0.668268i	$0.232964\pi$
$80$		0			0
$81$	−916.754	−	1587.86i	−1.25755	−	2.17814i
$82$		0			0
$83$	43.5791			0.0576317			0.0288158	−	0.999585i	$-0.490826\pi$
$83$	43.5791			0.0576317			0.0288158	+	0.999585i	$0.490826\pi$
$84$		0			0
$85$	544.041			0.694229
$86$		0			0
$87$	−401.956	−	696.209i	−0.495336	−	0.857947i
$88$		0			0
$89$	−196.547	+	340.429i	−0.234089	+	0.405454i	−0.959007	−	0.283381i	$-0.908544\pi$
$89$	−196.547	+	340.429i	−0.234089	+	0.405454i	0.724919	+	0.688834i	$0.241877\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−796.617	+	1379.78i	−0.888229	+	1.53846i
$94$		0			0
$95$	367.672	+	636.827i	0.397077	+	0.687758i
$96$		0			0
$97$	−521.748			−0.546139			−0.273070	−	0.961994i	$-0.588039\pi$
$97$	−521.748			−0.546139			−0.273070	+	0.961994i	$0.588039\pi$
$98$		0			0
$99$	−1210.40			−1.22878
$100$		0			0
$101$	−376.544	−	652.194i	−0.370966	−	0.642532i	0.618748	−	0.785589i	$-0.287640\pi$
$101$	−376.544	−	652.194i	−0.370966	−	0.642532i	−0.989714	+	0.143057i	$0.954307\pi$
$102$		0			0
$103$	465.461	−	806.202i	0.445274	−	0.771237i	−0.552797	−	0.833316i	$-0.686440\pi$
$103$	465.461	−	806.202i	0.445274	−	0.771237i	0.998071	+	0.0620787i	$0.0197730\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−841.521	+	1457.56i	−0.760308	+	1.31689i	0.182384	+	0.983227i	$0.441619\pi$
$107$	−841.521	+	1457.56i	−0.760308	+	1.31689i	−0.942692	+	0.333664i	$0.891715\pi$
$108$		0			0
$109$	391.953	+	678.882i	0.344424	+	0.596560i	0.985249	−	0.171127i	$-0.0547408\pi$
$109$	391.953	+	678.882i	0.344424	+	0.596560i	−0.640825	+	0.767687i	$0.721407\pi$
$110$		0			0
$111$	2997.15			2.56285
$112$		0			0
$113$	−628.205			−0.522979			−0.261489	−	0.965206i	$-0.584214\pi$
$113$	−628.205			−0.522979			−0.261489	+	0.965206i	$0.584214\pi$
$114$		0			0
$115$	192.469	+	333.366i	0.156068	+	0.270318i
$116$		0			0
$117$	−2997.50	+	5191.82i	−2.36854	+	4.10243i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	496.774	−	860.438i	0.373234	−	0.646460i
$122$		0			0
$123$	874.664	+	1514.96i	0.641186	+	1.11057i
$124$		0			0
$125$	1272.29			0.910377
$126$		0			0
$127$	328.466			0.229501			0.114751	−	0.993394i	$-0.463393\pi$
$127$	328.466			0.229501			0.114751	+	0.993394i	$0.463393\pi$
$128$		0			0
$129$	−396.939	−	687.519i	−0.270919	−	0.469245i
$130$		0			0
$131$	1206.40	−	2089.54i	0.804606	−	1.39362i	−0.111950	−	0.993714i	$-0.535710\pi$
$131$	1206.40	−	2089.54i	0.804606	−	1.39362i	0.916557	−	0.399905i	$-0.130957\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1109.18	−	1921.15i	0.707132	−	1.22479i
$136$		0			0
$137$	87.8493	+	152.159i	0.0547845	+	0.0948895i	0.892117	−	0.451804i	$-0.149219\pi$
$137$	87.8493	+	152.159i	0.0547845	+	0.0948895i	−0.837333	+	0.546694i	$0.815886\pi$
$138$		0			0
$139$	−1489.47			−0.908885			−0.454442	−	0.890776i	$-0.650161\pi$
$139$	−1489.47			−0.908885			−0.454442	+	0.890776i	$0.650161\pi$
$140$		0			0
$141$	−3545.89			−2.11786
$142$		0			0
$143$	835.685	+	1447.45i	0.488696	+	0.846446i
$144$		0			0
$145$	246.829	−	427.521i	0.141366	−	0.244853i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1517.16	+	2627.81i	−0.834168	+	1.44482i	0.0605388	+	0.998166i	$0.480718\pi$
$149$	−1517.16	+	2627.81i	−0.834168	+	1.44482i	−0.894706	+	0.446655i	$0.852615\pi$
$150$		0			0
$151$	1057.41	+	1831.49i	0.569873	+	0.987050i	0.996578	+	0.0826586i	$0.0263411\pi$
$151$	1057.41	+	1831.49i	0.569873	+	0.987050i	−0.426704	+	0.904391i	$0.640326\pi$
$152$		0			0
$153$	−6056.90			−3.20047
$154$		0			0
$155$	−978.356			−0.506990
$156$		0			0
$157$	−137.852	−	238.767i	−0.0700752	−	0.121374i	0.828859	−	0.559458i	$-0.188991\pi$
$157$	−137.852	−	238.767i	−0.0700752	−	0.121374i	−0.898934	+	0.438084i	$0.855657\pi$
$158$		0			0
$159$	−253.755	+	439.517i	−0.126567	+	0.219220i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−609.623	+	1055.90i	−0.292941	+	0.507389i	−0.974504	−	0.224371i	$-0.927967\pi$
$163$	−609.623	+	1055.90i	−0.292941	+	0.507389i	0.681563	+	0.731760i	$0.261301\pi$
$164$		0			0
$165$	−523.919	−	907.454i	−0.247194	−	0.428153i
$166$		0			0
$167$	−2781.99			−1.28908			−0.644542	−	0.764569i	$-0.722952\pi$
$167$	−2781.99			−1.28908			−0.644542	+	0.764569i	$0.722952\pi$
$168$		0			0
$169$	6081.15			2.76794
$170$		0			0
$171$	−4093.36	−	7089.91i	−1.83057	−	3.17064i
$172$		0			0
$173$	1353.52	−	2344.37i	0.594835	−	1.03029i	−0.398735	−	0.917066i	$-0.630551\pi$
$173$	1353.52	−	2344.37i	0.594835	−	1.03029i	0.993570	−	0.113219i	$-0.0361161\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2704.79	−	4684.83i	1.14861	−	1.98946i
$178$		0			0
$179$	860.863	+	1491.06i	0.359463	+	0.622608i	0.987871	−	0.155276i	$-0.0496265\pi$
$179$	860.863	+	1491.06i	0.359463	+	0.622608i	−0.628408	+	0.777884i	$0.716293\pi$
$180$		0			0
$181$	−624.549			−0.256477			−0.128238	−	0.991743i	$-0.540932\pi$
$181$	−624.549			−0.256477			−0.128238	+	0.991743i	$0.540932\pi$
$182$		0			0
$183$	−2663.01			−1.07571
$184$		0			0
$185$	920.228	+	1593.88i	0.365711	+	0.633430i
$186$		0			0
$187$	−844.314	+	1462.40i	−0.330173	+	0.571877i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2394.89	−	4148.07i	0.907269	−	1.57144i	0.0894257	−	0.995994i	$-0.471497\pi$
$191$	2394.89	−	4148.07i	0.907269	−	1.57144i	0.817843	−	0.575442i	$-0.195170\pi$
$192$		0			0
$193$	718.117	+	1243.81i	0.267830	+	0.463895i	0.968301	−	0.249786i	$-0.0803602\pi$
$193$	718.117	+	1243.81i	0.267830	+	0.463895i	−0.700471	+	0.713681i	$0.747027\pi$
$194$		0			0
$195$	−5189.85			−1.90591
$196$		0			0
$197$	2508.66			0.907282			0.453641	−	0.891184i	$-0.350125\pi$
$197$	2508.66			0.907282			0.453641	+	0.891184i	$0.350125\pi$
$198$		0			0
$199$	2046.88	+	3545.30i	0.729143	+	1.26291i	0.957246	+	0.289276i	$0.0934147\pi$
$199$	2046.88	+	3545.30i	0.729143	+	1.26291i	−0.228102	+	0.973637i	$0.573252\pi$
$200$		0			0
$201$	289.798	−	501.945i	0.101695	−	0.176142i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−537.105	+	930.292i	−0.182990	+	0.316948i
$206$		0			0
$207$	−2142.80	−	3711.43i	−0.719491	−	1.24619i
$208$		0			0
$209$	−2282.41			−0.755395
$210$		0			0
$211$	−2308.68			−0.753253			−0.376626	−	0.926365i	$-0.622916\pi$
$211$	−2308.68			−0.753253			−0.376626	+	0.926365i	$0.622916\pi$
$212$		0			0
$213$	749.515	+	1298.20i	0.241108	+	0.417611i
$214$		0			0
$215$	243.748	−	422.184i	0.0773185	−	0.133920i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	3524.98	−	6105.45i	1.08765	−	1.88387i
$220$		0			0
$221$	4181.82	+	7243.12i	1.27285	+	2.20464i
$222$		0			0
$223$	775.802			0.232967			0.116483	−	0.993193i	$-0.462838\pi$
$223$	775.802			0.232967			0.116483	+	0.993193i	$0.462838\pi$
$224$		0			0
$225$	−5928.35			−1.75655
$226$		0			0
$227$	−89.1192	−	154.359i	−0.0260575	−	0.0451329i	0.852703	−	0.522397i	$-0.174962\pi$
$227$	−89.1192	−	154.359i	−0.0260575	−	0.0451329i	−0.878760	+	0.477264i	$0.841629\pi$
$228$		0			0
$229$	−1304.91	+	2260.17i	−0.376553	+	0.652209i	−0.990558	−	0.137093i	$-0.956224\pi$
$229$	−1304.91	+	2260.17i	−0.376553	+	0.652209i	0.614005	+	0.789302i	$0.289557\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−787.919	+	1364.72i	−0.221538	+	0.383715i	−0.955275	−	0.295719i	$-0.904441\pi$
$233$	−787.919	+	1364.72i	−0.221538	+	0.383715i	0.733737	+	0.679433i	$0.237774\pi$
$234$		0			0
$235$	−1088.71	−	1885.70i	−0.302212	−	0.523446i
$236$		0			0
$237$	2798.71			0.767071
$238$		0			0
$239$	620.547			0.167949			0.0839746	−	0.996468i	$-0.473239\pi$
$239$	620.547			0.167949			0.0839746	+	0.996468i	$0.473239\pi$
$240$		0			0
$241$	−1276.40	−	2210.78i	−0.341161	−	0.590909i	0.643487	−	0.765457i	$-0.277487\pi$
$241$	−1276.40	−	2210.78i	−0.341161	−	0.590909i	−0.984649	+	0.174548i	$0.944154\pi$
$242$		0			0
$243$	−3775.51	+	6539.38i	−0.996704	+	1.72634i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5652.29	+	9790.05i	−1.45606	+	2.52197i
$248$		0			0
$249$	−210.007	−	363.743i	−0.0534484	−	0.0925753i
$250$		0			0
$251$	4057.62			1.02038			0.510189	−	0.860062i	$-0.329575\pi$
$251$	4057.62			1.02038			0.510189	+	0.860062i	$0.329575\pi$
$252$		0			0
$253$	−1194.80			−0.296902
$254$		0			0
$255$	−2621.72	−	4540.96i	−0.643838	−	1.11516i
$256$		0			0
$257$	2580.56	−	4469.67i	0.626347	−	1.08486i	−0.361932	−	0.932204i	$-0.617883\pi$
$257$	2580.56	−	4469.67i	0.626347	−	1.08486i	0.988279	−	0.152660i	$-0.0487839\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2747.99	+	4759.67i	−0.651711	+	1.12880i
$262$		0			0
$263$	−3559.82	−	6165.80i	−0.834632	−	1.44562i	−0.894330	−	0.447408i	$-0.852347\pi$
$263$	−3559.82	−	6165.80i	−0.834632	−	1.44562i	0.0596982	−	0.998216i	$-0.480986\pi$
$264$		0			0
$265$	−311.646			−0.0722426
$266$		0			0
$267$	3788.62			0.868389
$268$		0			0
$269$	−3001.35	−	5198.50i	−0.680282	−	1.17828i	−0.974895	−	0.222666i	$-0.928524\pi$
$269$	−3001.35	−	5198.50i	−0.680282	−	1.17828i	0.294613	−	0.955617i	$-0.404809\pi$
$270$		0			0
$271$	−396.542	+	686.831i	−0.0888864	+	0.153956i	−0.907041	−	0.421043i	$-0.861664\pi$
$271$	−396.542	+	686.831i	−0.0888864	+	0.153956i	0.818154	+	0.574999i	$0.194997\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−826.393	+	1431.36i	−0.181212	+	0.313869i
$276$		0			0
$277$	2027.17	+	3511.15i	0.439713	+	0.761606i	0.997667	−	0.0682662i	$-0.0217467\pi$
$277$	2027.17	+	3511.15i	0.439713	+	0.761606i	−0.557954	+	0.829872i	$0.688413\pi$
$278$		0			0
$279$	10892.2			2.33728
$280$		0			0
$281$	2990.91			0.634955			0.317478	−	0.948266i	$-0.397164\pi$
$281$	2990.91			0.634955			0.317478	+	0.948266i	$0.397164\pi$
$282$		0			0
$283$	2858.05	+	4950.30i	0.600331	+	1.03980i	0.992771	+	0.120026i	$0.0382979\pi$
$283$	2858.05	+	4950.30i	0.600331	+	1.03980i	−0.392440	+	0.919778i	$0.628369\pi$
$284$		0			0
$285$	3543.61	−	6137.71i	0.736510	−	1.27567i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1768.50	+	3063.13i	−0.359963	+	0.623475i
$290$		0			0
$291$	2514.30	+	4354.89i	0.506497	+	0.877279i
$292$		0			0
$293$	5301.47			1.05705			0.528524	−	0.848918i	$-0.322746\pi$
$293$	5301.47			1.05705			0.528524	+	0.848918i	$0.322746\pi$
$294$		0			0
$295$	3321.86			0.655613
$296$		0			0
$297$	3442.74	+	5963.00i	0.672619	+	1.16501i
$298$		0			0
$299$	−2958.86	+	5124.90i	−0.572293	+	0.991240i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−3629.12	+	6285.83i	−0.688078	+	1.19179i
$304$		0			0
$305$	−817.638	−	1416.19i	−0.153501	−	0.265872i
$306$		0			0
$307$	−5978.40			−1.11142			−0.555709	−	0.831377i	$-0.687553\pi$
$307$	−5978.40			−1.11142			−0.555709	+	0.831377i	$0.687553\pi$
$308$		0			0
$309$	−8972.19			−1.65181
$310$		0			0
$311$	2425.36	+	4200.84i	0.442217	+	0.765942i	0.997854	−	0.0654832i	$-0.0208589\pi$
$311$	2425.36	+	4200.84i	0.442217	+	0.765942i	−0.555637	+	0.831425i	$0.687526\pi$
$312$		0			0
$313$	−2227.67	+	3858.44i	−0.402285	+	0.696779i	−0.994001	−	0.109368i	$-0.965117\pi$
$313$	−2227.67	+	3858.44i	−0.402285	+	0.696779i	0.591716	+	0.806147i	$0.298451\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5130.93	−	8887.04i	0.909091	−	1.57459i	0.0937624	−	0.995595i	$-0.470111\pi$
$317$	5130.93	−	8887.04i	0.909091	−	1.57459i	0.815329	−	0.578998i	$-0.196556\pi$
$318$		0			0
$319$	766.124	+	1326.97i	0.134466	+	0.232902i
$320$		0			0
$321$	16221.1			2.82048
$322$		0			0
$323$	−11421.3			−1.96749
$324$		0			0
$325$	4093.06	+	7089.38i	0.698591	+	1.20999i
$326$		0			0
$327$	3777.63	−	6543.04i	0.638848	−	1.10652i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1543.56	−	2673.52i	0.256319	−	0.443958i	−0.708934	−	0.705275i	$-0.750824\pi$
$331$	1543.56	−	2673.52i	0.256319	−	0.443958i	0.965253	+	0.261317i	$0.0841569\pi$
$332$		0			0
$333$	−10245.1	−	17745.0i	−1.68596	−	2.92018i
$334$		0			0
$335$	355.912			0.0580465
$336$		0			0
$337$	4588.57			0.741707			0.370853	−	0.928691i	$-0.379065\pi$
$337$	4588.57			0.741707			0.370853	+	0.928691i	$0.379065\pi$
$338$		0			0
$339$	3027.31	+	5243.46i	0.485018	+	0.840075i
$340$		0			0
$341$	1518.34	−	2629.85i	0.241123	−	0.417637i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1855.01	−	3212.98i	0.289480	−	0.501394i
$346$		0			0
$347$	4822.79	+	8353.32i	0.746112	+	1.29230i	0.949673	+	0.313242i	$0.101415\pi$
$347$	4822.79	+	8353.32i	0.746112	+	1.29230i	−0.203561	+	0.979062i	$0.565252\pi$
$348$		0			0
$349$	−2612.70			−0.400730			−0.200365	−	0.979721i	$-0.564213\pi$
$349$	−2612.70			−0.400730			−0.200365	+	0.979721i	$0.564213\pi$
$350$		0			0
$351$	34103.2			5.18602
$352$		0			0
$353$	−4344.04	−	7524.09i	−0.654985	−	1.13447i	−0.981897	−	0.189413i	$-0.939341\pi$
$353$	−4344.04	−	7524.09i	−0.654985	−	1.13447i	0.326912	−	0.945055i	$-0.393992\pi$
$354$		0			0
$355$	−460.254	+	797.184i	−0.0688106	+	0.119183i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1369.41	−	2371.89i	0.201322	−	0.348701i	−0.747632	−	0.664113i	$-0.768809\pi$
$359$	1369.41	−	2371.89i	0.201322	−	0.348701i	0.948955	+	0.315412i	$0.102143\pi$
$360$		0			0
$361$	−4289.22	−	7429.14i	−0.625342	−	1.08312i
$362$		0			0
$363$	−9575.79			−1.38457
$364$		0			0
$365$	4329.17			0.620819
$366$		0			0
$367$	−984.294	−	1704.85i	−0.139999	−	0.242486i	0.787497	−	0.616319i	$-0.211377\pi$
$367$	−984.294	−	1704.85i	−0.139999	−	0.242486i	−0.927496	+	0.373833i	$0.878043\pi$
$368$		0			0
$369$	5979.69	−	10357.1i	0.843604	−	1.46117i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4957.80	−	8587.16i	0.688217	−	1.19203i	−0.284197	−	0.958766i	$-0.591727\pi$
$373$	4957.80	−	8587.16i	0.688217	−	1.19203i	0.972414	−	0.233262i	$-0.0749398\pi$
$374$		0			0
$375$	−6131.15	−	10619.5i	−0.844296	−	1.46236i
$376$		0			0
$377$	7589.10			1.03676
$378$		0			0
$379$	10933.4			1.48183			0.740913	−	0.671601i	$-0.234393\pi$
$379$	10933.4			1.48183			0.740913	+	0.671601i	$0.234393\pi$
$380$		0			0
$381$	−1582.87	−	2741.62i	−0.212843	−	0.368654i
$382$		0			0
$383$	6004.84	−	10400.7i	0.801131	−	1.38760i	−0.117742	−	0.993044i	$-0.537565\pi$
$383$	6004.84	−	10400.7i	0.801131	−	1.38760i	0.918872	−	0.394555i	$-0.129101\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2713.69	+	4700.25i	−0.356446	+	0.617383i
$388$		0			0
$389$	−4621.13	−	8004.04i	−0.602316	−	1.04324i	−0.992470	−	0.122492i	$-0.960912\pi$
$389$	−4621.13	−	8004.04i	−0.602316	−	1.04324i	0.390154	−	0.920750i	$-0.372422\pi$
$390$		0			0
$391$	−5978.84			−0.773306
$392$		0			0
$393$	−23254.4			−2.98481
$394$		0			0
$395$	859.301	+	1488.35i	0.109459	+	0.189588i
$396$		0			0
$397$	−1851.23	+	3206.42i	−0.234032	+	0.405354i	−0.958991	−	0.283437i	$-0.908525\pi$
$397$	−1851.23	+	3206.42i	−0.234032	+	0.405354i	0.724959	+	0.688792i	$0.241859\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1597.39	+	2766.76i	−0.198927	+	0.344552i	−0.948181	−	0.317731i	$-0.897079\pi$
$401$	−1597.39	+	2766.76i	−0.198927	+	0.344552i	0.749254	+	0.662283i	$0.230412\pi$
$402$		0			0
$403$	−7520.22	−	13025.4i	−0.929551	−	1.61003i
$404$		0			0
$405$	−10851.4			−1.33138
$406$		0			0
$407$	−5712.52			−0.695723
$408$		0			0
$409$	−3108.79	−	5384.59i	−0.375843	−	0.650980i	0.614610	−	0.788831i	$-0.289314\pi$
$409$	−3108.79	−	5384.59i	−0.375843	−	0.650980i	−0.990453	+	0.137852i	$0.955980\pi$
$410$		0			0
$411$	846.689	−	1466.51i	0.101616	−	0.176004i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	128.959	−	223.363i	0.0152538	−	0.0264204i
$416$		0			0
$417$	7177.72	+	12432.2i	0.842913	+	1.45997i
$418$		0			0
$419$	−290.002			−0.0338127			−0.0169063	−	0.999857i	$-0.505382\pi$
$419$	−290.002			−0.0338127			−0.0169063	+	0.999857i	$0.505382\pi$
$420$		0			0
$421$	16109.3			1.86489			0.932445	−	0.361311i	$-0.117671\pi$
$421$	16109.3			1.86489			0.932445	+	0.361311i	$0.117671\pi$
$422$		0			0
$423$	12120.8	+	20993.9i	1.39323	+	2.41314i
$424$		0			0
$425$	−4135.32	+	7162.59i	−0.471983	+	0.817498i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	8054.30	−	13950.5i	0.906446	−	1.57001i
$430$		0			0
$431$	35.6998	+	61.8339i	0.00398979	+	0.00691052i	0.868013	−	0.496541i	$-0.165397\pi$
$431$	35.6998	+	61.8339i	0.00398979	+	0.00691052i	−0.864024	+	0.503451i	$0.832063\pi$
$432$		0			0
$433$	−2473.42			−0.274515			−0.137258	−	0.990535i	$-0.543829\pi$
$433$	−2473.42			−0.274515			−0.137258	+	0.990535i	$0.543829\pi$
$434$		0			0
$435$	−4757.86			−0.524418
$436$		0			0
$437$	−4040.60	−	6998.53i	−0.442307	−	0.766098i
$438$		0			0
$439$	−6525.92	+	11303.2i	−0.709488	+	1.22887i	0.255560	+	0.966793i	$0.417740\pi$
$439$	−6525.92	+	11303.2i	−0.709488	+	1.22887i	−0.965047	+	0.262076i	$0.915593\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2552.18	+	4420.50i	−0.273719	+	0.474096i	−0.969811	−	0.243857i	$-0.921587\pi$
$443$	−2552.18	+	4420.50i	−0.273719	+	0.474096i	0.696092	+	0.717953i	$0.254921\pi$
$444$		0			0
$445$	1163.24	+	2014.79i	0.123916	+	0.214629i
$446$		0			0
$447$	29244.8			3.09447
$448$		0			0
$449$	−7106.77			−0.746969			−0.373485	−	0.927636i	$-0.621837\pi$
$449$	−7106.77			−0.746969			−0.373485	+	0.927636i	$0.621837\pi$
$450$		0			0
$451$	−1667.10	−	2887.50i	−0.174059	−	0.301479i
$452$		0			0
$453$	10191.3	−	17651.8i	1.05702	−	1.83081i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−471.474	+	816.618i	−0.0482596	+	0.0835881i	−0.889146	−	0.457623i	$-0.848701\pi$
$457$	−471.474	+	816.618i	−0.0482596	+	0.0835881i	0.840887	+	0.541211i	$0.182034\pi$
$458$		0			0
$459$	17227.7	+	29839.2i	1.75189	+	3.03437i
$460$		0			0
$461$	−1906.71			−0.192634			−0.0963172	−	0.995351i	$-0.530706\pi$
$461$	−1906.71			−0.192634			−0.0963172	+	0.995351i	$0.530706\pi$
$462$		0			0
$463$	−4896.83			−0.491523			−0.245761	−	0.969330i	$-0.579038\pi$
$463$	−4896.83			−0.491523			−0.245761	+	0.969330i	$0.579038\pi$
$464$		0			0
$465$	4714.68	+	8166.07i	0.470190	+	0.814392i
$466$		0			0
$467$	−181.948	+	315.143i	−0.0180290	+	0.0312271i	−0.874899	−	0.484305i	$-0.839072\pi$
$467$	−181.948	+	315.143i	−0.0180290	+	0.0312271i	0.856870	+	0.515532i	$0.172406\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1328.62	+	2301.23i	−0.129977	+	0.225128i
$472$		0			0
$473$	756.561	+	1310.40i	0.0735449	+	0.127383i
$474$		0			0
$475$	−11178.9			−1.07984
$476$		0			0
$477$	3469.62			0.333046
$478$		0			0
$479$	2611.92	+	4523.98i	0.249148	+	0.431537i	0.963290	−	0.268464i	$-0.0865162\pi$
$479$	2611.92	+	4523.98i	0.249148	+	0.431537i	−0.714142	+	0.700001i	$0.753183\pi$
$480$		0			0
$481$	−14146.8	+	24503.0i	−1.34104	+	2.32275i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1543.95	+	2674.20i	−0.144551	+	0.250370i
$486$		0			0
$487$	−6066.58	−	10507.6i	−0.564482	−	0.977712i	−0.997098	−	0.0761334i	$-0.975743\pi$
$487$	−6066.58	−	10507.6i	−0.564482	−	0.977712i	0.432615	−	0.901579i	$-0.357591\pi$
$488$		0			0
$489$	11751.1			1.08671
$490$		0			0
$491$	2875.76			0.264320			0.132160	−	0.991228i	$-0.457809\pi$
$491$	2875.76			0.264320			0.132160	+	0.991228i	$0.457809\pi$
$492$		0			0
$493$	3833.73	+	6640.22i	0.350228	+	0.606613i
$494$		0			0
$495$	−3581.80	+	6203.85i	−0.325232	+	0.563318i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−7821.93	+	13548.0i	−0.701719	+	1.21541i	0.266143	+	0.963933i	$0.414251\pi$
$499$	−7821.93	+	13548.0i	−0.701719	+	1.21541i	−0.967863	+	0.251480i	$0.919083\pi$
$500$		0			0
$501$	13406.4	+	23220.5i	1.19551	+	2.07069i
$502$		0			0
$503$	−3530.58			−0.312964			−0.156482	−	0.987681i	$-0.550015\pi$
$503$	−3530.58			−0.312964			−0.156482	+	0.987681i	$0.550015\pi$
$504$		0			0
$505$	−4457.07			−0.392746
$506$		0			0
$507$	−29305.0	−	50757.7i	−2.56702	−	4.44621i
$508$		0			0
$509$	3026.80	−	5242.57i	0.263577	−	0.456528i	−0.703613	−	0.710583i	$-0.748431\pi$
$509$	3026.80	−	5242.57i	0.263577	−	0.456528i	0.967190	+	0.254055i	$0.0817645\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−23285.5	+	40331.7i	−2.00406	+	3.47113i
$514$		0			0
$515$	−2754.77	−	4771.41i	−0.235708	−	0.408259i
$516$		0			0
$517$	6758.43			0.574923
$518$		0			0
$519$	−26090.4			−2.20663
$520$		0			0
$521$	−10438.2	−	18079.5i	−0.877746	−	1.52030i	−0.853808	−	0.520588i	$-0.825713\pi$
$521$	−10438.2	−	18079.5i	−0.877746	−	1.52030i	−0.0239383	−	0.999713i	$-0.507621\pi$
$522$		0			0
$523$	−9508.68	+	16469.5i	−0.795001	+	1.37698i	0.127837	+	0.991795i	$0.459196\pi$
$523$	−9508.68	+	16469.5i	−0.795001	+	1.37698i	−0.922838	+	0.385187i	$0.874137\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7597.88	−	13159.9i	0.628024	−	1.08777i
$528$		0			0
$529$	3968.32	+	6873.34i	0.326155	+	0.564916i
$530$		0			0
$531$	−36982.9			−3.02245
$532$		0			0
$533$	−16514.0			−1.34203
$534$		0			0
$535$	4980.44	+	8626.38i	0.402474	+	0.697105i
$536$		0			0
$537$	8296.97	−	14370.8i	0.666742	−	1.15483i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7035.33	+	12185.6i	−0.559099	+	0.968388i	0.438473	+	0.898744i	$0.355519\pi$
$541$	−7035.33	+	12185.6i	−0.559099	+	0.968388i	−0.997572	+	0.0696433i	$0.977814\pi$
$542$		0			0
$543$	3009.69	+	5212.94i	0.237860	+	0.411986i
$544$		0			0
$545$	4639.45			0.364646
$546$		0			0
$547$	−16099.5			−1.25844			−0.629218	−	0.777229i	$-0.716625\pi$
$547$	−16099.5			−1.25844			−0.629218	+	0.777229i	$0.716625\pi$
$548$		0			0
$549$	9102.91	+	15766.7i	0.707656	+	1.22570i
$550$		0			0
$551$	−5181.80	+	8975.15i	−0.400639	+	0.693928i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	8869.12	−	15361.8i	0.678330	−	1.17490i
$556$		0			0
$557$	−6163.61	−	10675.7i	−0.468870	−	0.812107i	0.530497	−	0.847687i	$-0.322005\pi$
$557$	−6163.61	−	10675.7i	−0.468870	−	0.812107i	−0.999367	+	0.0355800i	$0.988672\pi$
$558$		0			0
$559$	7494.37			0.567045
$560$		0			0
$561$	16274.9			1.22483
$562$		0			0
$563$	2246.17	+	3890.48i	0.168143	+	0.291233i	0.937767	−	0.347265i	$-0.112890\pi$
$563$	2246.17	+	3890.48i	0.168143	+	0.291233i	−0.769624	+	0.638498i	$0.779556\pi$
$564$		0			0
$565$	−1858.98	+	3219.85i	−0.138421	+	0.239752i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8447.56	−	14631.6i	0.622390	−	1.07801i	−0.366649	−	0.930359i	$-0.619495\pi$
$569$	8447.56	−	14631.6i	0.622390	−	1.07801i	0.989039	−	0.147652i	$-0.0471716\pi$
$570$		0			0
$571$	44.0576	+	76.3099i	0.00322899	+	0.00559277i	0.867635	−	0.497201i	$-0.165639\pi$
$571$	44.0576	+	76.3099i	0.00322899	+	0.00559277i	−0.864406	+	0.502794i	$0.832306\pi$
$572$		0			0
$573$	−46163.8			−3.36565
$574$		0			0
$575$	−5851.93			−0.424422
$576$		0			0
$577$	4086.46	+	7077.96i	0.294838	+	0.510674i	0.974947	−	0.222436i	$-0.0714010\pi$
$577$	4086.46	+	7077.96i	0.294838	+	0.510674i	−0.680109	+	0.733111i	$0.738068\pi$
$578$		0			0
$579$	6921.18	−	11987.8i	0.496778	−	0.860445i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	483.654	−	837.714i	0.0343583	−	0.0595104i
$584$		0			0
$585$	17740.3	+	30727.2i	1.25380	+	2.17165i
$586$		0			0
$587$	18301.4			1.28685			0.643425	−	0.765509i	$-0.277513\pi$
$587$	18301.4			1.28685			0.643425	+	0.765509i	$0.277513\pi$
$588$		0			0
$589$	20539.1			1.43684
$590$		0			0
$591$	−12089.2	−	20939.1i	−0.841426	−	1.45739i
$592$		0			0
$593$	−1034.59	+	1791.97i	−0.0716452	+	0.124093i	−0.899622	−	0.436669i	$-0.856158\pi$
$593$	−1034.59	+	1791.97i	−0.0716452	+	0.124093i	0.827977	+	0.560762i	$0.189492\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19727.8	−	34169.5i	1.35244	−	2.34249i
$598$		0			0
$599$	−6807.81	−	11791.5i	−0.464373	−	0.804318i	0.534800	−	0.844979i	$-0.320387\pi$
$599$	−6807.81	−	11791.5i	−0.464373	−	0.804318i	−0.999173	+	0.0406606i	$0.987054\pi$
$600$		0			0
$601$	1041.12			0.0706624			0.0353312	−	0.999376i	$-0.488751\pi$
$601$	1041.12			0.0706624			0.0353312	+	0.999376i	$0.488751\pi$
$602$		0			0
$603$	−3962.44			−0.267600
$604$		0			0
$605$	−2940.10	−	5092.40i	−0.197574	−	0.342207i
$606$		0			0
$607$	−7747.98	+	13419.9i	−0.518090	+	0.897358i	0.481689	+	0.876342i	$0.340023\pi$
$607$	−7747.98	+	13419.9i	−0.518090	+	0.897358i	−0.999779	+	0.0210160i	$0.993310\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16737.0	−	28989.3i	1.10819	−	1.91944i
$612$		0			0
$613$	−468.229	−	810.996i	−0.0308509	−	0.0534353i	0.850188	−	0.526480i	$-0.176488\pi$
$613$	−468.229	−	810.996i	−0.0308509	−	0.0534353i	−0.881039	+	0.473044i	$0.843155\pi$
$614$		0			0
$615$	10353.2			0.678831
$616$		0			0
$617$	−798.142			−0.0520778			−0.0260389	−	0.999661i	$-0.508289\pi$
$617$	−798.142			−0.0520778			−0.0260389	+	0.999661i	$0.508289\pi$
$618$		0			0
$619$	−499.517	−	865.189i	−0.0324350	−	0.0561791i	0.849352	−	0.527826i	$-0.176993\pi$
$619$	−499.517	−	865.189i	−0.0324350	−	0.0561791i	−0.881787	+	0.471647i	$0.843660\pi$
$620$		0			0
$621$	−12189.5	+	21112.9i	−0.787679	+	1.36430i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1858.35	+	3218.75i	−0.118934	+	0.206000i
$626$		0			0
$627$	10998.9	+	19050.6i	0.700564	+	1.21341i
$628$		0			0
$629$	−28585.8			−1.81207
$630$		0			0
$631$	25618.7			1.61626			0.808132	−	0.589001i	$-0.200479\pi$
$631$	25618.7			1.61626			0.808132	+	0.589001i	$0.200479\pi$
$632$		0			0
$633$	11125.5	+	19270.0i	0.698577	+	1.20997i
$634$		0			0
$635$	971.994	−	1683.54i	0.0607439	−	0.105212i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	5124.10	−	8875.20i	0.317224	−	0.549448i
$640$		0			0
$641$	−9175.68	−	15892.7i	−0.565394	−	0.979291i	−0.997013	−	0.0772350i	$-0.975391\pi$
$641$	−9175.68	−	15892.7i	−0.565394	−	0.979291i	0.431619	−	0.902056i	$-0.357943\pi$
$642$		0			0
$643$	−22068.9			−1.35352			−0.676759	−	0.736205i	$-0.736616\pi$
$643$	−22068.9			−1.35352			−0.676759	+	0.736205i	$0.736616\pi$
$644$		0			0
$645$	−4698.47			−0.286825
$646$		0			0
$647$	3204.75	+	5550.79i	0.194732	+	0.337286i	0.946813	−	0.321785i	$-0.104283\pi$
$647$	3204.75	+	5550.79i	0.194732	+	0.337286i	−0.752081	+	0.659071i	$0.770950\pi$
$648$		0			0
$649$	−5155.30	+	8929.24i	−0.311808	+	0.540067i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10549.4	+	18272.1i	−0.632206	+	1.09501i	0.354894	+	0.934907i	$0.384517\pi$
$653$	−10549.4	+	18272.1i	−0.632206	+	1.09501i	−0.987100	+	0.160106i	$0.948816\pi$
$654$		0			0
$655$	−7139.92	−	12366.7i	−0.425923	−	0.737721i
$656$		0			0
$657$	−48197.4			−2.86204
$658$		0			0
$659$	−29509.0			−1.74432			−0.872159	−	0.489222i	$-0.837281\pi$
$659$	−29509.0			−1.74432			−0.872159	+	0.489222i	$0.837281\pi$
$660$		0			0
$661$	−10858.9	−	18808.2i	−0.638976	−	1.10674i	−0.985658	−	0.168756i	$-0.946025\pi$
$661$	−10858.9	−	18808.2i	−0.638976	−	1.10674i	0.346682	−	0.937983i	$-0.387308\pi$
$662$		0			0
$663$	40304.2	−	69809.0i	2.36091	−	4.08922i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2712.58	+	4698.32i	−0.157468	+	0.272743i
$668$		0			0
$669$	−3738.58	−	6475.41i	−0.216057	−	0.374221i
$670$		0			0
$671$	5075.67			0.292018
$672$		0			0
$673$	3168.78			0.181497			0.0907486	−	0.995874i	$-0.471074\pi$
$673$	3168.78			0.181497			0.0907486	+	0.995874i	$0.471074\pi$
$674$		0			0
$675$	16862.0	+	29205.9i	0.961510	+	1.66538i
$676$		0			0
$677$	11293.4	−	19560.8i	0.641124	−	1.11046i	−0.344058	−	0.938948i	$-0.611802\pi$
$677$	11293.4	−	19560.8i	0.641124	−	1.11046i	0.985182	−	0.171511i	$-0.0548650\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−858.928	+	1487.71i	−0.0483321	+	0.0837137i
$682$		0			0
$683$	3565.41	+	6175.47i	0.199746	+	0.345970i	0.948446	−	0.316939i	$-0.102655\pi$
$683$	3565.41	+	6175.47i	0.199746	+	0.345970i	−0.748700	+	0.662909i	$0.769322\pi$
$684$		0			0
$685$	1039.85			0.0580010
$686$		0			0
$687$	25153.3			1.39688
$688$		0			0
$689$	−2395.50	−	4149.13i	−0.132455	−	0.229418i
$690$		0			0
$691$	5707.28	−	9885.29i	0.314204	−	0.544217i	−0.665064	−	0.746786i	$-0.731596\pi$
$691$	5707.28	−	9885.29i	0.314204	−	0.544217i	0.979268	+	0.202569i	$0.0649291\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4407.62	+	7634.22i	−0.240562	+	0.416665i
$696$		0			0
$697$	−8342.27	−	14449.2i	−0.453352	−	0.785228i
$698$		0			0
$699$	15187.9			0.821829
$700$		0			0
$701$	10005.8			0.539106			0.269553	−	0.962986i	$-0.413124\pi$
$701$	10005.8			0.539106			0.269553	+	0.962986i	$0.413124\pi$
$702$		0			0
$703$	−19318.8	−	33461.1i	−1.03645	−	1.79518i
$704$		0			0
$705$	−10493.0	+	18174.4i	−0.560551	+	0.970902i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4770.10	−	8262.06i	0.252673	−	0.437642i	−0.711588	−	0.702597i	$-0.752024\pi$
$709$	4770.10	−	8262.06i	0.252673	−	0.437642i	0.964261	+	0.264955i	$0.0853571\pi$
$710$		0			0
$711$	−9566.76	−	16570.1i	−0.504615	−	0.874019i
$712$		0			0
$713$	10751.8			0.564739
$714$		0			0
$715$	9891.80			0.517388
$716$		0			0
$717$	−2990.41	−	5179.54i	−0.155758	−	0.269782i
$718$		0			0
$719$	−2856.77	+	4948.07i	−0.148177	+	0.256651i	−0.930554	−	0.366155i	$-0.880674\pi$
$719$	−2856.77	+	4948.07i	−0.148177	+	0.256651i	0.782377	+	0.622806i	$0.214007\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−12301.9	+	21307.5i	−0.632796	+	1.09603i
$724$		0			0
$725$	3752.36	+	6499.28i	0.192220	+	0.332934i
$726$		0			0
$727$	−18615.1			−0.949649			−0.474824	−	0.880081i	$-0.657488\pi$
$727$	−18615.1			−0.949649			−0.474824	+	0.880081i	$0.657488\pi$
$728$		0			0
$729$	23271.8			1.18233
$730$		0			0
$731$	3785.88	+	6557.33i	0.191554	+	0.331781i
$732$		0			0
$733$	−13226.1	+	22908.3i	−0.666463	+	1.15435i	0.312424	+	0.949943i	$0.398859\pi$
$733$	−13226.1	+	22908.3i	−0.666463	+	1.15435i	−0.978887	+	0.204404i	$0.934474\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−552.352	+	956.702i	−0.0276067	+	0.0478162i
$738$		0			0
$739$	11458.0	+	19845.9i	0.570353	+	0.987880i	0.996530	+	0.0832398i	$0.0265267\pi$
$739$	11458.0	+	19845.9i	0.570353	+	0.987880i	−0.426177	+	0.904640i	$0.640140\pi$
$740$		0			0
$741$	108953.			5.40148
$742$		0			0
$743$	−24313.3			−1.20050			−0.600248	−	0.799814i	$-0.704931\pi$
$743$	−24313.3			−1.20050			−0.600248	+	0.799814i	$0.704931\pi$
$744$		0			0
$745$	8979.16	+	15552.4i	0.441572	+	0.764825i
$746$		0			0
$747$	−1435.72	+	2486.74i	−0.0703218	+	0.121801i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2809.24	−	4865.74i	0.136499	−	0.236423i	−0.789670	−	0.613531i	$-0.789748\pi$
$751$	2809.24	−	4865.74i	0.136499	−	0.236423i	0.926169	+	0.377109i	$0.123082\pi$
$752$		0			0
$753$	−19553.6	−	33867.9i	−0.946313	−	1.63906i
$754$		0			0
$755$	12516.3			0.603332
$756$		0			0
$757$	−2786.08			−0.133767			−0.0668837	−	0.997761i	$-0.521306\pi$
$757$	−2786.08			−0.133767			−0.0668837	+	0.997761i	$0.521306\pi$
$758$		0			0
$759$	5757.71	+	9972.64i	0.275351	+	0.476922i
$760$		0			0
$761$	3407.22	−	5901.47i	0.162302	−	0.281115i	−0.773392	−	0.633928i	$-0.781442\pi$
$761$	3407.22	−	5901.47i	0.162302	−	0.281115i	0.935694	+	0.352813i	$0.114775\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−17923.5	+	31044.5i	−0.847094	+	1.46721i
$766$		0			0
$767$	25533.8	+	44225.8i	1.20205	+	2.08201i
$768$		0			0
$769$	13274.3			0.622474			0.311237	−	0.950332i	$-0.399257\pi$
$769$	13274.3			0.622474			0.311237	+	0.950332i	$0.399257\pi$
$770$		0			0
$771$	−49742.8			−2.32353
$772$		0			0
$773$	9325.70	+	16152.6i	0.433922	+	0.751576i	0.997207	−	0.0746872i	$-0.0237958\pi$
$773$	9325.70	+	16152.6i	0.433922	+	0.751576i	−0.563285	+	0.826263i	$0.690463\pi$
$774$		0			0
$775$	7436.61	−	12880.6i	0.344685	−	0.597012i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11275.7	−	19530.1i	0.518606	−	0.898252i
$780$		0			0
$781$	−1428.57	−	2474.35i	−0.0654522	−	0.113367i
$782$		0			0
$783$	31264.5			1.42695
$784$		0			0
$785$	−1631.72			−0.0741895
$786$		0			0
$787$	17514.2	+	30335.5i	0.793284	+	1.37401i	0.923923	+	0.382578i	$0.124964\pi$
$787$	17514.2	+	30335.5i	0.793284	+	1.37401i	−0.130639	+	0.991430i	$0.541703\pi$
$788$		0			0
$789$	−34309.5	+	59425.7i	−1.54810	+	2.68138i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	12569.7	−	21771.4i	0.562879	−	0.974935i
$794$		0			0
$795$	1501.82	+	2601.23i	0.0669988	+	0.116045i
$796$		0			0
$797$	−1170.81			−0.0520352			−0.0260176	−	0.999661i	$-0.508283\pi$
$797$	−1170.81			−0.0520352			−0.0260176	+	0.999661i	$0.508283\pi$
$798$		0			0
$799$	33819.6			1.49744
$800$		0			0
$801$	−12950.5	−	22431.0i	−0.571267	−	0.989463i
$802$		0			0
$803$	−6718.57	+	11636.9i	−0.295259	+	0.511404i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−28927.0	+	50103.0i	−1.26181	+	2.18551i
$808$		0			0
$809$	−6677.38	−	11565.6i	−0.290190	−	0.502625i	0.683664	−	0.729797i	$-0.260385\pi$
$809$	−6677.38	−	11565.6i	−0.290190	−	0.502625i	−0.973855	+	0.227172i	$0.927052\pi$
$810$		0			0
$811$	12327.2			0.533745			0.266872	−	0.963732i	$-0.414010\pi$
$811$	12327.2			0.533745			0.266872	+	0.963732i	$0.414010\pi$
$812$		0			0
$813$	7643.72			0.329738
$814$		0			0
$815$	3607.98	+	6249.21i	0.155070	+	0.268589i
$816$		0			0
$817$	−5117.12	+	8863.12i	−0.219125	+	0.379536i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22726.9	+	39364.2i	−0.966109	+	1.67335i	−0.259504	+	0.965742i	$0.583559\pi$
$821$	−22726.9	+	39364.2i	−0.966109	+	1.67335i	−0.706605	+	0.707608i	$0.749774\pi$
$822$		0			0
$823$	−4643.12	−	8042.13i	−0.196657	−	0.340621i	0.750785	−	0.660547i	$-0.229675\pi$
$823$	−4643.12	−	8042.13i	−0.196657	−	0.340621i	−0.947443	+	0.319926i	$0.896342\pi$
$824$		0			0
$825$	15929.5			0.672235
$826$		0			0
$827$	22224.4			0.934485			0.467242	−	0.884129i	$-0.345248\pi$
$827$	22224.4			0.934485			0.467242	+	0.884129i	$0.345248\pi$
$828$		0			0
$829$	22018.4	+	38136.9i	0.922473	+	1.59777i	0.795576	+	0.605853i	$0.207168\pi$
$829$	22018.4	+	38136.9i	0.922473	+	1.59777i	0.126896	+	0.991916i	$0.459499\pi$
$830$		0			0
$831$	19537.8	−	33840.4i	0.815593	−	1.41265i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−8232.44	+	14259.0i	−0.341192	+	0.590962i
$836$		0			0
$837$	−30980.8	−	53660.3i	−1.27939	−	2.21597i
$838$		0			0
$839$	37521.9			1.54398			0.771991	−	0.635633i	$-0.219261\pi$
$839$	37521.9			1.54398			0.771991	+	0.635633i	$0.219261\pi$
$840$		0			0
$841$	−17431.6			−0.714732
$842$		0			0
$843$	−14413.1	−	24964.3i	−0.588867	−	1.01995i
$844$		0			0
$845$	17995.3	−	31168.8i	0.732612	−	1.26892i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	27545.8	−	47710.8i	1.11351	−	1.92866i
$850$		0			0
$851$	−10113.0	−	17516.3i	−0.407367	−	0.705581i
$852$		0			0
$853$	−34441.3			−1.38247			−0.691235	−	0.722630i	$-0.742933\pi$
$853$	−34441.3			−1.38247			−0.691235	+	0.722630i	$0.742933\pi$
$854$		0			0
$855$	−48452.1			−1.93804
$856$		0			0
$857$	−12485.4	−	21625.4i	−0.497659	−	0.861970i	0.502338	−	0.864672i	$-0.332473\pi$
$857$	−12485.4	−	21625.4i	−0.497659	−	0.861970i	−0.999996	+	0.00270129i	$0.999140\pi$
$858$		0			0
$859$	8677.77	−	15030.3i	0.344682	−	0.597006i	−0.640614	−	0.767863i	$-0.721320\pi$
$859$	8677.77	−	15030.3i	0.344682	−	0.597006i	0.985296	+	0.170857i	$0.0546535\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13284.0	+	23008.5i	−0.523977	+	0.907555i	0.475633	+	0.879644i	$0.342219\pi$
$863$	−13284.0	+	23008.5i	−0.523977	+	0.907555i	−0.999610	+	0.0279113i	$0.991114\pi$
$864$		0			0
$865$	−8010.67	−	13874.9i	−0.314880	−	0.545388i
$866$		0			0
$867$	34089.5			1.33534
$868$		0			0
$869$	−5334.31			−0.208232
$870$		0			0
$871$	2735.75	+	4738.46i	0.106426	+	0.184336i
$872$		0			0
$873$	17189.1	−	29772.4i	0.666395	−	1.15423i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23617.7	+	40907.0i	−0.909363	+	1.57506i	−0.0944124	+	0.995533i	$0.530097\pi$
$877$	−23617.7	+	40907.0i	−0.909363	+	1.57506i	−0.814951	+	0.579530i	$0.803236\pi$
$878$		0			0
$879$	−25547.7	−	44249.9i	−0.980321	−	1.69797i
$880$		0			0
$881$	−29598.0			−1.13188			−0.565938	−	0.824448i	$-0.691486\pi$
$881$	−29598.0			−1.13188			−0.565938	+	0.824448i	$0.691486\pi$
$882$		0			0
$883$	42952.5			1.63700			0.818498	−	0.574509i	$-0.194807\pi$
$883$	42952.5			1.63700			0.818498	+	0.574509i	$0.194807\pi$
$884$		0			0
$885$	−16008.0	−	27726.6i	−0.608025	−	1.05313i
$886$		0			0
$887$	−16748.9	+	29010.0i	−0.634018	+	1.09815i	0.352704	+	0.935735i	$0.385262\pi$
$887$	−16748.9	+	29010.0i	−0.634018	+	1.09815i	−0.986722	+	0.162417i	$0.948071\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	16840.6	−	29168.8i	0.633202	−	1.09674i
$892$		0			0
$893$	22855.9	+	39587.5i	0.856486	+	1.48348i
$894$		0			0
$895$	10189.8			0.380568
$896$		0			0
$897$	57034.9			2.12301
$898$		0			0
$899$	−6894.26	−	11941.2i	−0.255769	−	0.443005i
$900$		0			0
$901$	2420.24	−	4191.97i	0.0894891	−	0.155000i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1848.16	+	3201.10i	−0.0678838	+	0.117578i
$906$		0			0
$907$	−6875.57	−	11908.8i	−0.251708	−	0.435972i	0.712288	−	0.701887i	$-0.247659\pi$
$907$	−6875.57	−	11908.8i	−0.251708	−	0.435972i	−0.963996	+	0.265916i	$0.914326\pi$
$908$		0			0
$909$	49621.4			1.81060
$910$		0			0
$911$	38706.9			1.40770			0.703851	−	0.710348i	$-0.251462\pi$
$911$	38706.9			1.40770			0.703851	+	0.710348i	$0.251462\pi$
$912$		0			0
$913$	400.271	+	693.289i	0.0145093	+	0.0251309i
$914$		0			0
$915$	−7880.37	+	13649.2i	−0.284718	+	0.493146i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	11646.2	−	20171.8i	0.418034	−	0.724056i	−0.577707	−	0.816244i	$-0.696053\pi$
$919$	11646.2	−	20171.8i	0.418034	−	0.724056i	0.995742	+	0.0921874i	$0.0293859\pi$
$920$		0			0
$921$	28809.8	+	49900.0i	1.03074	+	1.78530i
$922$		0			0
$923$	−14151.1			−0.504649
$924$		0			0
$925$	−27979.1			−0.994537
$926$		0			0
$927$	30669.4	+	53121.0i	1.08664	+	1.88212i
$928$		0			0
$929$	21462.7	−	37174.4i	0.757984	−	1.31287i	−0.185893	−	0.982570i	$-0.559518\pi$
$929$	21462.7	−	37174.4i	0.757984	−	1.31287i	0.943877	−	0.330297i	$-0.107149\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	23375.5	−	40487.6i	0.820236	−	1.42069i
$934$		0			0
$935$	4996.97	+	8655.01i	0.174779	+	0.302726i
$936$		0			0
$937$	−44869.6			−1.56438			−0.782191	−	0.623039i	$-0.785898\pi$
$937$	−44869.6			−1.56438			−0.782191	+	0.623039i	$0.785898\pi$
$938$		0			0
$939$	42940.4			1.49234
$940$		0			0
$941$	−6121.69	−	10603.1i	−0.212074	−	0.367323i	0.740290	−	0.672288i	$-0.234688\pi$
$941$	−6121.69	−	10603.1i	−0.212074	−	0.367323i	−0.952363	+	0.304966i	$0.901355\pi$
$942$		0			0
$943$	5902.61	−	10223.6i	0.203834	−	0.353051i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21013.5	−	36396.5i	0.721065	−	1.24892i	−0.239509	−	0.970894i	$-0.576986\pi$
$947$	21013.5	−	36396.5i	0.721065	−	1.24892i	0.960573	−	0.278027i	$-0.0896803\pi$
$948$		0			0
$949$	33276.5	+	57636.6i	1.13825	+	1.97151i
$950$		0			0
$951$	−98903.6			−3.37242
$952$		0			0
$953$	15757.3			0.535601			0.267801	−	0.963474i	$-0.413703\pi$
$953$	15757.3			0.535601			0.267801	+	0.963474i	$0.413703\pi$
$954$		0			0
$955$	−14173.9	−	24549.9i	−0.480268	−	0.831849i
$956$		0			0
$957$	7383.88	−	12789.3i	0.249412	−	0.431994i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1232.13	−	2134.10i	0.0413590	−	0.0716359i
$962$		0			0
$963$	−55448.2	−	96039.1i	−1.85544	−	3.21372i
$964$		0			0
$965$	8500.17			0.283555
$966$		0			0
$967$	−25863.4			−0.860093			−0.430046	−	0.902807i	$-0.641503\pi$
$967$	−25863.4			−0.860093			−0.430046	+	0.902807i	$0.641503\pi$
$968$		0			0
$969$	55039.1	+	95330.5i	1.82468	+	3.16043i
$970$		0			0
$971$	23922.8	−	41435.5i	0.790647	−	1.36944i	−0.134919	−	0.990857i	$-0.543077\pi$
$971$	23922.8	−	41435.5i	0.790647	−	1.36944i	0.925567	−	0.378585i	$-0.123589\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	39448.8	−	68327.3i	1.29577	−	2.24433i
$976$		0			0
$977$	−27361.3	−	47391.2i	−0.895973	−	1.55187i	−0.832595	−	0.553882i	$-0.813146\pi$
$977$	−27361.3	−	47391.2i	−0.895973	−	1.55187i	−0.0633779	−	0.997990i	$-0.520187\pi$
$978$		0			0
$979$	−7221.06			−0.235737
$980$		0			0
$981$	−51651.8			−1.68106
$982$		0			0
$983$	15031.3	+	26035.0i	0.487715	+	0.844747i	0.999900	−	0.0141280i	$-0.00449724\pi$
$983$	15031.3	+	26035.0i	0.487715	+	0.844747i	−0.512185	+	0.858875i	$0.671164\pi$
$984$		0			0
$985$	7423.60	−	12858.1i	0.240138	−	0.415931i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2678.72	+	4639.67i	−0.0861256	+	0.149174i
$990$		0			0
$991$	18643.3	+	32291.2i	0.597603	+	1.03508i	0.993174	+	0.116643i	$0.0372133\pi$
$991$	18643.3	+	32291.2i	0.597603	+	1.03508i	−0.395571	+	0.918435i	$0.629453\pi$
$992$		0			0
$993$	−29753.5			−0.950855
$994$		0			0
$995$	24228.4			0.771953
$996$		0			0
$997$	29467.2	+	51038.7i	0.936045	+	1.62128i	0.772760	+	0.634698i	$0.218876\pi$
$997$	29467.2	+	51038.7i	0.936045	+	1.62128i	0.163285	+	0.986579i	$0.447791\pi$
$998$		0			0
$999$	−58280.2	+	100944.i	−1.84575	+	3.19693i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.4.i.o.177.1		8
7.2	even	3		392.4.a.n.1.4	yes	4
7.3	odd	6	inner	392.4.i.o.361.4		8
7.4	even	3	inner	392.4.i.o.361.1		8
7.5	odd	6		392.4.a.n.1.1	✓	4
7.6	odd	2	inner	392.4.i.o.177.4		8
28.19	even	6		784.4.a.bh.1.4		4
28.23	odd	6		784.4.a.bh.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.4.a.n.1.1	✓	4	7.5	odd	6
392.4.a.n.1.4	yes	4	7.2	even	3
392.4.i.o.177.1		8	1.1	even	1	trivial
392.4.i.o.177.4		8	7.6	odd	2	inner
392.4.i.o.361.1		8	7.4	even	3	inner
392.4.i.o.361.4		8	7.3	odd	6	inner
784.4.a.bh.1.1		4	28.23	odd	6
784.4.a.bh.1.4		4	28.19	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	392.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.81898 - 8.34673i) q^{3} +(2.95919 - 5.12547i) q^{5} +(-32.9452 + 57.0628i) q^{9} +O(q^{10})\)	\(q+(-4.81898 - 8.34673i) q^{3} +(2.95919 - 5.12547i) q^{5} +(-32.9452 + 57.0628i) q^{9} +(9.18493 + 15.9088i) q^{11} +90.9844 q^{13} -57.0412 q^{15} +(45.9619 + 79.6084i) q^{17} +(-62.1237 + 107.601i) q^{19} +(-32.5206 + 56.3273i) q^{23} +(44.9864 + 77.9187i) q^{25} +374.825 q^{27} +83.4110 q^{29} +(-82.6540 - 143.161i) q^{31} +(88.5240 - 153.328i) q^{33} +(-155.486 + 269.310i) q^{37} +(-438.452 - 759.421i) q^{39} -181.504 q^{41} +82.3699 q^{43} +(194.982 + 337.719i) q^{45} +(183.954 - 318.618i) q^{47} +(442.980 - 767.263i) q^{51} +(-26.3287 - 45.6026i) q^{53} +108.720 q^{55} +1197.49 q^{57} +(280.639 + 486.081i) q^{59} +(138.152 - 239.287i) q^{61} +(269.240 - 466.337i) q^{65} +(30.0684 + 52.0800i) q^{67} +626.865 q^{69} -155.534 q^{71} +(365.739 + 633.479i) q^{73} +(433.577 - 750.978i) q^{75} +(-145.192 + 251.480i) q^{79} +(-916.754 - 1587.86i) q^{81} +43.5791 q^{83} +544.041 q^{85} +(-401.956 - 696.209i) q^{87} +(-196.547 + 340.429i) q^{89} +(-796.617 + 1379.78i) q^{93} +(367.672 + 636.827i) q^{95} -521.748 q^{97} -1210.40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 136 q^{9}+O(q^{10}) \) 8 * q - 136 * q^9	\( 8 q - 136 q^{9} + 116 q^{11} + 224 q^{15} + 80 q^{23} - 448 q^{25} + 72 q^{29} - 436 q^{37} - 2232 q^{39} + 744 q^{43} + 780 q^{51} - 976 q^{53} + 7624 q^{57} - 780 q^{65} + 1176 q^{67} - 4816 q^{71} - 56 q^{79} - 2104 q^{81} + 9880 q^{85} - 2376 q^{93} - 4032 q^{95} - 5176 q^{99}+O(q^{100}) \) 8 * q - 136 * q^9 + 116 * q^11 + 224 * q^15 + 80 * q^23 - 448 * q^25 + 72 * q^29 - 436 * q^37 - 2232 * q^39 + 744 * q^43 + 780 * q^51 - 976 * q^53 + 7624 * q^57 - 780 * q^65 + 1176 * q^67 - 4816 * q^71 - 56 * q^79 - 2104 * q^81 + 9880 * q^85 - 2376 * q^93 - 4032 * q^95 - 5176 * q^99

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